A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always $8m$ .Find the equation of the path traced by him if the distance between the poles is $6m$.

End points of Latus Rectum $(\pm \large\frac{b^2}{a}$$,ae)$ and $(\pm \large\frac{b^2}{a}$$,-ae)$

Directrices $y=\pm \large\frac{a}{e}$.

The major axis is $x=0$ (y- axis) and the minor axes is $y=0$ (x- axis)

If a point moves so that the sum of its distances from two fixed points is a constant,then the path traced as an ellipse with major axis of length equal to the constant sum and foci at the two fixed points.

Step 1:

Since the sum of his distances from the two poles is a constant (8 m) the player is tracing an ellipse with the two poles as foci.

The sum of the distances =$FP+F'P=2a=8$

Therefore $a=4m$

Also $FF'$=distance between the poles =$2ae$

$2ae=6\Rightarrow 2\times 4e=6$

$\Rightarrow e=\large\frac{3}{4}$

Step 2:

The path traced is an ellipse with $a=4,e=\large\frac{3}{4}$

Therefore $b=a\sqrt{1-e^2}$

$\Rightarrow 4\sqrt{1-\large\frac{9}{16}}=\sqrt 7$

The equation of the ellipse is $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$